Warm up A remote-controlled car's wheel accelerates at 22.4 rad/s 2. If the wheel begins with an angular speed of 10.8 rad/s, what is the wheel's angular speed after exactly three full turns? AP Physics Torque Torque rotational force which causes a change in rotational motion. This force is defined by linear force multiplied by a radius. Fr Unit is Newton meter Nm r lever arm Perpendicular to force through center of rotation 1
Torque is a vector quantity. 125 N is applied to a nut by a wrench. The length of the wrench is 0.300 m. Find the torque =? Fr 2
a) A mechanics tightens the lugs on a tire by applying a torque of 100.N m at an angle of 90 to the line of action. What force is applied if the wrench is 0.40 m long? b) How long must the wrench be if the mechanic is only capable of applying a force of 200N? If the Force is Not Perpendicular: Torque is: rf sin Push on door as shown. Calculate the torque? (Top view) 1.5 m 55.0 330 N 3
Fr sin 330 N 1.5 m sin 55.0 410 Nm Torque Practice: Find the torque about the axis indicated by the black circle. 25 cm 35 N 40 4
10 cm 40 N 40 Practice page 5 rotation packet 5
Equilibrium and Torque: If object is in rotational equilibrium, net torque about any axis is zero. = 0 Two Conditions for Static Equilibrium: F = 0 = 0 Sum of external forces = 0 Sum of external torques = 0 CW clockwise rotation around center point CCW counterclockwise rotation around center point 6
Two Torques Acting at same time: Clockwise Rotation negative torque Counter Clockwise Rotation positive torque = 1 + 2 = F1d1 + F2d2 Teeter totter is in equilibrium as shown: = 0 1 = F 1 r 1 and 2 = F 2 r 2 and 1 = 2, then 7
Fr F 11 F 1 r 1 = F 2 r 2 2 r 2 r 2 385 N 1.55 m 225 N r 2 = 2.65 m A 50.0 N see-saw supports two people who weigh 455 N and 525 N. The fulcrum is under the CG of the board. The 525 N person is 1.50 m from the center. Where should the smaller person sit so it balances? Find the upward force n exerted by fulcrum on the board. 1.50 m x 8
n 1.50 m x 50.0 N 525 N 455 N Fy 0 n 525 N 455 N 50.0 N 0 n 525 N 455 N 50.0 N n=1030n y 0 1 2 Fr F r 1 1 2 2 r 2 Fr F 11 2 r 2 525 N 1.50 455 N m r2 1.73 m 9
A 530-N woman is ready to back flip off the 3.9 m diving board. The board has negligible weight. a) Draw a force diagram showing all the forces. b) Use torque summation to solve for the force of the fulcrum on the board. c) Use force summation to solve for the force of the bolt on the board. 10
A uniform 1500 kg beam 20.0m long, supports a 15,000kg printing press 5.0 m from the right support column. Calculate the force on each of the vertical support columns. Balance Beam A 150 kg balance beam is supported at each end. A 65 kg gymnast stands a third of the way from side B. What is the vertical force (Fb) on the support closest to the gymnast? A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs 205 N and is 3.00 m long. Assume the 675 N worker stands 1.00 m from the left 11
end of the scaffold. What is the force on the rope farther from the worker? Page 6 Rotation packet 12
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8.5 Rotational Dynamics; Torque and Rotational Inertia -Inertia mass (translational motion) is an objects ability to resist a change in motion. -Moment of inertia (I) is an objects resistance to rotational acceleration (ɑ). This resistance is dependent on: 1. the amount of mass present in the object, and 2. the distribution of that mass about the chosen axis of rotation. -Objects with smaller moments of inertia are easier to accelerate I = mr 2 14
Equation sheet for different objects. Figure 8-21 gives moments of inertia or various object of uniform composition. You will be given this info on the exam Find the moment of inertia of two 5-kg masses joined by a meter long rod of negligible mass, when rotated about the center of the rod. Compare this to the moment of inertia when rotated about one of its masses. 15
Moment of Inertia Find the moment of inertia of the following objects: M=2 kg M=2 kg 30 cm M=6 kg 80 cm R=40 cm Solid Sphere M=2 kg Hoop Mass=4 kg 50 cm R= 20 cm Log Mass=50 kg 2 m 16
Warm up -Draw a free body diagram of an object rolling down a ramp 17
Newton s second law for rotation We saw in our study of dynamics that forces cause acceleration: ΣF= ma Torques produce angular acceleration, and the rotational equivalent of mass is the moment of inertia, I: Στ= Iα 18
Sample Problem 1- Mel spins a top with a moment of inertia of 0.001kg m 2 on a table by applying a torque of 0.01N m for two seconds. If the top starts from rest, find the angular velocity of the top. Sample Problem 2- What is the angular acceleration experienced by a uniform solid disc of mass 2-kg and radius 0.1m when a net torque of 10N m is applied? Assume the disc spins about its center. 19
will happen to? Sample Problem 3-A 6 Newton force is applied at the edge of a solid disk with a radius of 0.6 meters and a rotational inertia of 4 kgm 2. a.what is the angular acceleration of the disk? b.if F doubles, what will happen to? c.if the rotational inertia doubles, what Sample Problem 4- The rotational inertia of a solid disk can be found using the equation I = ½ MR 2. A force of 100 Newtons is applied to a solid disk with a mass of 10 kg, and a radius of.5 meters. Starting from rest, how long will it take to reach an angular velocity of 4 radians/sec? 20
Sample problem 5 - A disk with a rotational inertia of 4 kg m 2, and a radius of 0.8 meters is attached to a wall by a pin through its center, and is free to rotate without friction. A string is wound around the disk, and attached to a 2 kg mass, as shown. The mass is dropped, causing the wheel to spin. Find the acceleration of the mass, and the rotational acceleration of the disk. 21
Sample problem 6- Use the diagram below to answer the questions: a. Draw a free-body diagram, and write an F net equation for the mass. b. Draw a rotational free-body diagram, and write a net equation for the wheel. c. Find the acceleration of the mass, the rotational acceleration of the wheel, and the tension in the string. Linear momentum-how hard something is to stop. p=mv momentum is conserved Angular momentum (L) how difficult to stop an object from rotating. Product of an objects moment of inertia (I) and its angular velocity (w). Is conserved. L=mvr sinθ Derive L for a point particle moving in a circle 22
Angular Momentum of a Single Particle Let s take the example of a tetherball of mass m =10 kg swinging about on a rope of length r = 2m and an angular velocity of 1 rad/s. Find the angular momentum of the tetherball. (Magnitude and direction) Previously we saw that the net force acting on an object is equal to the rate of change of the object s momentum with time. Similarly, the net torque acting on an object is equal to the rate of change of the object s angular momentum with time: τ net = ΔL Δt 23
If the net torque action on a rigid body is zero, then the angular momentum of the body is constant or conserved. Angular velocity increases as r decreases. DEMO A light rigid rod 1.00 m in length rotates in the xy plane about a pivot through the rod's center. Two particles of masses 4.00 kg and 3.00 kg are connected to its ends. Determine the angular momentum (magnitude and direction) of the system about the origin at the time the speed of each particle is 5.00 m/s. 24
A playground merry-go-round of radius R = 2.0 m has a moment of inertia I = 250 kg m 2 and is rotating at 10 rev/min. A 25-kg child jumps onto the edge of the merry-go-round. What is the new angular speed of the merry-go-round? 25